Italian

Verb

A figurate number is a number that can be
represented as a regular and discrete geometric pattern (e.g. dots).
If the pattern is polytopic, the figurate is
labeled a polytopic number, and may be a polygonal
number or a polyhedral number.

The first few triangular
numbers can be built from rows of 1, 2, 3, 4, 5, and 6 items:
The n-th regular r-topic number is given by the formula:

Gnomon

Figurate numbers were a concern of Pythagorean
geometry, since Pythagoras is credited with initiating them,
and the notion that these numbers are generated from a gnomon or
basic unit. The gnomon is the piece added to a figurate number to
transform it to the next bigger one.

For example, the gnomon of the square number is
the odd
number, of the general form 2n + 1, n = 0, 1, 2, 3, ... . The
square of size 8 composed of gnomons looks like this:

To transform from the n-square (the square of
size n) to the (n + 1)-square, one adjoins 2n + 1 elements: one to
the end of each row (n elements), one to the end of each column (n
elements), and a single one to the corner. For example, when
transforming the 7-square to the 8-square, we add 15 elements;
these adjunctions are the 8s in the above figure.

The tedium of increasing number of subtractions
as the number grows is bypassed by a method similar to the standard
way of square-rooting taught in school. For example: 1225 = 35 ×
35, Note the sum of the digits of this square root: 3 + 5 = 8. This
square-root shortcut reduces 35 subtractions to only 8
subtractions. The shortcut involves two "tricks": a markoff trick,
and resumptive trick.

The markoff trick is already known from the
familiar square root algorithm. One marks off the target number in
pairs of digits, from the right, as in marking 1225 as 12′25; then,
calculation begins with the first digit-pair to the left. The
reason is that squaring a one-digit number results in a 1- or
2-digit square. Thus, 1, 2, 3 have, respectively, the 1-digit
squares of 1, 4, 9. But 4 has the 2-digit square of 16; and numbers
5, 6, 7, 8, 9 have 2-digit squares. To allow for this, one begins
with two digits to provide one digit at each process-stage.

The resumptive trick (unique to this present
algorithm) shifts from one pair of target number digits to its next
(rightward) two digits, explained in calculating the square root of
1225.

Mark off 1225 as 12′25; begin calculation with left pair of
digits, namely, 12.

The last "successful" subtrahend was 5; but the next odd
number, 7, cannot be subtracted, so interpolate between 5 and 7 for
number 6. (This is "first part" of the resumptive trick.)

Since (noted above) the successful 3 subtractions actually
represent the 2-digit 30, treat the interpolated 6 as 60; resume
odd number subtraction with the first odd number in the sixties,
namely, 61. (This is "second and final part" of the resumptive
trick or subalgorithm.)

Having passed from 325 to 0 by five subtractions, the second
digit is 5: and 30 + 5 = 35, that is, the square root of 1225 is
35, obtained in exactly 3 + 5 = 8 subtractions by applying the
markoff and resumptive tricks or subalgorithms.

This procedure (taking many words to explain, but
quickly executed) is not restricted to calculating square roots of
natural numbers or positive integers. It can even be applied toward
calculating the irrational square root of 2, to any number of
decimal places.

Demonstration of mathematical properties

School children
construct figurate numbers from pebbles, bottle caps, etc. As a
bonus, children can use figurate numbers to discover the commutative
law and associative
law for addition
and multiplication — laws
usually dictated to them — by building rows and tables of
dots.

For example, the additive commutativity of 2 + 3
= 3 + 2 = 5 becomes:

And the multiplicative commutativity of 2 × 3 = 3
× 2 = 6 becomes:

Besides the subtractive method, the additive
method can also approximate square roots of positive integers and
solve quadratic equations.

The concepts of figurate numbers and gnomon
implicitly anticipate the modern concept of recursion.